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13
votes
3answers
346 views

Infiniteness of the Galois cohomology over a number field with coefficients in a finite Galois module

Let $k$ be a number field and $M$ be a nonzero finite discrete $\mathrm{Gal}(\bar k/k)$-module. Is it true that $H^1(k,M)$ is infinite? This would complete the answer of Daniel Loughran. There is a ...
11
votes
1answer
400 views

Elements of arbitrary large order in the first Galois cohomology of an elliptic curve

Let $E$ be an elliptic curve over $k=\mathbb{Q}$. Consider $H^1(k,E)$. In this answer Daniel Loughran writes: "I'm pretty sure that this cohomology group has elements of arbitrarily large order". I ...
7
votes
1answer
253 views

Weight filtration on certain Galois representations

Let $G$ be the absolute Galois group of a number field $K$. Let $\ell$ be a prime number. There are representations $\mathbb{Z}_\ell(n)$ of $G$ on the group of $\ell$-adic integers given by the ...
6
votes
2answers
394 views

First Galois cohomology of Weil restriction of $\mathbb{G}_m$

Let $L/K$ be a finite Galois extension, write $G:= Gal(L/K)$. Denote by $R = Res(\mathbb{G}_m)$ the Weil restriction of $\mathbb{G}_m$, from $L$ to $K$. I want to show that its first Galois cohomology ...
3
votes
1answer
310 views

Galois cohomology of a non-abelian group over a function field

Let $k$ be an algebraically closed field, and $X$ a connected smooth projective curve over $X$. Let $F$ be the function field of $k$. Let $G$ be an algebraic group over $k$ (assume that it is smooth, ...
5
votes
1answer
175 views

Applications of the Galois embedding problem

Given a finite Galois extension of number fields $L/K$ with Galois group $G$ and a surjection $E\twoheadrightarrow G$ of finite groups, the Galois embedding problem is the question of whether there ...
2
votes
1answer
93 views

F-points of product of closed subgroups vs. product of F-points, F a local field, reference?

Let $F$ be a finite extension of $\mathbb Q_p$, where p is an odd prime. Let $G$ be a connected reductive group defined over $F$. Let $M, H$ be closed $F$-subgroups of $G$ (in particular, I'm ...
7
votes
2answers
454 views

Galois cohomologies of an elliptic curve

I asked this question at math stackexchange but did not get any answer and I was suggested to post the question here. I am studying basic theory of elliptic curves. I encountered Galois cohomology. ...
4
votes
0answers
91 views

Norm variety for n=5, p=2 not isomorphic to a quadric

In the paper "Motivic construction of cohomological invariants", the author displays a list of known norm varieties for several $n,p$ on page $11$. For $p=2, n=5$ it says that a norm variety is given ...
7
votes
1answer
312 views

A vanishing condition for cup products in Galois cohomology

Let $k$ be a field of characteristic $\neq 2$. For a non-zero element $a \in k^*$, let us write $[a] \in H^1(k,\mathbb{Z}/2)$ for the Galois cohomology class corresponding to the quadratic extension ...
2
votes
0answers
263 views

Generic triviality of $G$-bundles

Let $k$ be an algebraically closed field and $X$ a curve over $k$. Then any $G$-bundle on a curve (where $G$ is reductive and connected) is generically trivial. This is the one of the main results of ...
1
vote
0answers
69 views

Motivic Pfister type varieties and norm varieties

Due to results of Rost it is known that the Grothendieck-Chow motiv of a Pfister quadric $X$ belonging to a pure $\alpha \in H^n(k,\mu_2)$ is decomposable in the following way $M(X) = ...
9
votes
1answer
242 views

Torsors trivializing over a fixed finite etale cover

Let $S$ be an integral regular scheme and let $T\to S$ be a finite etale morphism. Let $G$ be a smooth affine finite type group scheme over $S$. Is the set of $S$-isomorphism classes of $G$-torsors ...
4
votes
1answer
238 views

Lifting torsors in characteristic $p$ to characteristic zero

Let $R$ be a local integral domain with residue field $k$ such that $R$ is of characteristic zero and $k$ is of characteristic $p>0$. Let $G$ be a smooth finite type affine group scheme with ...
2
votes
2answers
376 views

Lifting projective Galois representation with condition

Let $\bar{\rho}: G_K\to PGL_n(\mathbb{C})$ be projective representation of the absolute Galois group of a number field $K$ and $\varphi\in Aut(G_K)$. A theorem of Tate tells us that we can always ...
3
votes
1answer
137 views

Classification of 3-forms in dimension 7

I'm looking for a classification of $3$-forms over a real vector space of dimension $7$ as for the $3$-forms in dimension $6$. References on the latter case are R. Bryant On the geometry of almost ...
1
vote
0answers
80 views

explicit zero 2-cocycle

Let $G$ be a group which acts linearly on a vector space of dimension $n$ over a field $k$. Denote by $\rho$ this representation and consider the associated adjoint representation $Ad\rho$ which is ...
3
votes
1answer
250 views

A question on the cohomology of elliptic curves over local fields

Let $K$ be a number field,$\nu$ a nonarchimedian prime of $K$, $K_{\nu} $ the completion of $K $ at $\nu $ with maximal unramified extension $K_{\nu}^{unr} $. Let $E $ be an elliptic curve defined ...
5
votes
1answer
266 views

Motives of a variety of type D4

Over the last decade Nikita Semenov, Skip Garibaldi and others have made some progress in the theory of cohomological invariants, (Rost)-motives and motivic decompositions of algebraic groups. For ...
4
votes
1answer
182 views

Twists of projective automorphisms

Let $X$ be a projective variety over a perfect field $k$. Recall that a twist of $X$ is a variety $Y$ over $k$ such that $$X_{\bar k} \cong Y_{\bar k}.$$ The twists of $X$ are classified by the Galois ...
1
vote
0answers
79 views

Local duality for abelian varieties

Let $A$ be an abelian variety over a p-adic field $K$. Let $I$ be the inertia group of $K$. There is a Yoneda pairing $$H^n(\hat{\mathbb{Z}},A^I) \times Ext^{2-n}_{\hat{\mathbb{Z}}}(A^I,\mathbb{Z}) ...
2
votes
1answer
339 views

Exactness on rational points of algebraic groups

Let $k$ be a finite extension of the p-adic number field $Q_p$ and G be a connected algebraic (not affine) group over $k$. It is well-known (see e.g. [1] Proposition 3.1) that G decomposes as ...
2
votes
1answer
214 views

extensions of crystalline representations

Denote by $G_p$ a choice of an absolute Galois group of $Q_p$, the field of $p$-adic numbers. Consider a continuous representations of $G_p$ on a $3$-dimensional $Q_p$ vector space that is a ...
8
votes
1answer
299 views

When does the continuous Galois(=etale) cohomology of fields coincide with the naive one? Often true by the Bloch-Kato conjecture?

For a field $F$ I am interested in its $l$-adic (Galois=\'etale) cohomology; here $l$ is a prime distinct from the characteristic of $F$ (for simplicity one may assume that the latter is $0$). For ...
1
vote
1answer
273 views

Splitting varieties of two Galois cohomology symbols

One characteristic property of the so called norm varieties defined by Rost, is that they split pure symbols of Galois cohomology groups $H^n(k,\mu_p)$, meaning: For some $\alpha \in H^n(k,\mu_p)$ ...
3
votes
1answer
123 views

On Serre's problem regarding the injectivity of Albert-Algebra cohomological invariants

In these Lecture Notes http://molle.fernuni-hagen.de/~loos/jordan/archive/cohinv/cohinv.pdf from 2006 by Garibaldi on page 21. 7.5 there is the following open problem mentioned: Is the map $g_3 ...
1
vote
1answer
132 views

Cohomological dimension of transcendental p-adic extensions

Let $q$ denote a quadratic form over a field $k$. The u-invariant of a field $u(k)$ is defined by $u(k):=\{ max (\mathrm{rank}(q)) $ | $ q $ is anisotropic over $k\}$. Let $k = \mathbb{Q}_p$ for any ...
8
votes
2answers
390 views

“Forms” of quadrics

The theory of Severi-Brauer varieties is well-known. Let $k$ be a (perfect) field. There may exist varieties not isomorphic to $\mathbf{P}^n$ over $k$, which are isomorphic to $\mathbf{P}^n$ over ...
1
vote
0answers
98 views

What is classified by $H^1(\mathbb{R},SO(p,q))$ and by $H^1(\mathbb{R},SU(p,q))$?

We denote by $F^{\mathbb{R}}_{p,q}$ the quadratic form over the field ${\mathbb{R}}$ $$ F^{\mathbb{R}}_{p,q}(x)=x_1^2+\dots+x_p^2-(x_{p+1}^2+\dots+x_{p+q}^2) $$ on the vector space ...
3
votes
1answer
269 views

The cardinality of first non-abelian Galois cohomology

Let $G$ be a linear algebraic group over a non-archimedean local field $F$. Let $H^1(F,G)$ be the first non-abelian Galois cohomology. It is known that when $F$ is of characteristic 0, i.e. finite ...
5
votes
1answer
187 views

Rank four quadratic Form with non trivial discriminant in I(k)

Im sure this is a beginners question. Let $k$ be a field and $I(k)$ the fundamental ideal in the Witt-ring W(k). The Arason-Pfister-Hauptsatz states: "If $\varphi$ is any anisotropic class in ...
0
votes
0answers
108 views

torsors on quasi-split groups

Let $\mathbf{G}$ be a split connected reductive group scheme over a scheme $X$. Let $X'\rightarrow X$ an étale Galois cover of group $\Gamma$. We consider $G$ a quasi-split group scheme over $X$ ...
5
votes
1answer
143 views

$H^1$ and fractional ideals group

Let $L/K$ be a Galois extension with Galois group and $\mathfrak p$ be a prime of the ring of integers $\mathcal O_K$. I would like to prove that $H^1(G, I_{\mathfrak p})=1$ where $I_{\mathfrak p}$ is ...
8
votes
1answer
422 views

Variant of Hilbert 90 for Galois extensions

Let $K/\mathbb F_q(x)$ be a finite Galois extension with Galois group $G$. Let $Aut(K)$ be the group of $\mathbb F_q$-automorphisms of $K$. Obviously, $G\subseteq Aut(K)$. It is well known that ...
15
votes
1answer
366 views

Computability of Brauer groups

A friend of mine and I were talking about computable algebra, and this question came up. The answer may already be known, but I couldn't find it with Google: Suppose I have a countable field, $k$. ...
6
votes
1answer
309 views

Continuity of l-adic cohomology: is the cohomology of the generic point isomorphic to the completion of the limit of cohomology of open subvarieties?

Let $X$ be a variety over an algebraically closed field $k$. Denote by $\eta$ its generic point; it is the inverse limit of the open subvarieties $X_i$ of $X$. It is well known that the etale ...
7
votes
1answer
1k views

Relations between the cohomology of discrete groups and of profinite groups

Let $G$ be a discrete group and $K$ be the profinite completion of $G$. Let $C_K$ denote the category of contionuous $K$-modules and ${C_K}'$ denotes category of finite continuous $K$-modules. Now for ...
5
votes
1answer
289 views

rationality question while dealing with an isogeny

I don't think that the following is known, but before going to other things, I would like to know what can be said about it. Thanks in advance for any relevant comment ! So here is the situation. Let ...
7
votes
2answers
405 views

Forms of algebraic varieties

Let $X$ be an algebraic variety (say, projective, irreducible and smooth), defined over a field $K$, and let $L$ be a Galois extension. I am interested in algebraic varieties $Y$, defined over $K$, ...
5
votes
1answer
291 views

Open subgroups of the etale fundamental group of $P^1_\mathbb Q\setminus\{0,\infty\}$

Let $G$ be the etale fundamental group of $P^1_\mathbb Q\setminus\{0,\infty\}$. Then $G$ is isomorphic to a semidirect product of $\widehat {\mathbb Z}(1)$ by $ Gal_\mathbb Q$. Is it true that ...
10
votes
0answers
408 views

Unramified Galois cohomology of number fields

I'm trying to understand unramified Galois cohomology of number fields a bit better. Set-up: Let $k$ be a number field and $S$ a finite set of places of $k$ which contains all the archimedean ...
4
votes
1answer
207 views

What is the interpretation of this galois cohomology set?

Let $K$ be a field of characteristic zero. Let $G_K:=Gal(\bar{K}/K)$ The nontrivial elements of the set $H^1(G_K,PGL_2)$ correspond to $\bar{K}/K$-forms of $\mathbb{P}^1$; i.e. curves that are ...
6
votes
0answers
116 views

References for Gauss Composition using Galois Cohomology

Note: I have already posted this on stackexchange, but have not yet gotten a response. What are some good references for the Gauss composition law on binary quadratic forms in terms of Galois ...
1
vote
1answer
259 views

Cohomology after completion

I've been scouring google and asking friend about something I was certain must be absolutely the easiest thing to people who do homological algebra, and none seem to know the answer to this, so if ...
1
vote
1answer
159 views

$L/k$ forms for affine schemes of finite type

Notations and terminology: Let $k$ be a field and $X$ be a $k$-scheme. Denote by $X_L$ the scheme $X\times_k\rm Spec(L)$. For a field extension $L/k$, a $L/k$ form is a $k$-scheme $Y$ such that there ...
0
votes
0answers
180 views

on the Galois cohomology of reductive groups

Let $G$ a simply connected group over an algebraically closed field. $F=k((t))$ and $\mathcal{O}=k[[t]]$. Let $\gamma\in G(\mathcal{O})\cap G(F)^{rs}$. Let $E=k((t^{1/n}))$ with $n$ prime to the ...
3
votes
1answer
90 views

Set of isomorphisms of Pfister forms corresponding to first cohomology of algebraic group

Assume $k_0$ is a field with char($k_0$) not $2$. Let us define functors from $\rm Field_{/k_0}\to \rm Sets$ as $\rm Pfister_n(k):=\{\text{isomorphism classes of n-fold Pfister forms over k}\}$; ...
0
votes
0answers
176 views

Descent for group actions

Suppose I have a finite Galois extension of fields $K/k$, as well as a finite group $G$ with a surjection $f: G \rightarrow \mathrm{Gal}(K/k)$. Finally, suppose I have an action $\sigma$ of $G$ on a ...
6
votes
2answers
684 views

Peu or très ramifiée extension

Let $p$ be a prime number. Let $\mathbb{F}$ be a finite extension of $\mathbb{F}_p$. Let $\omega$ be the mod $p$ cyclotomic character and let $V$ be a representation of $G_{p} = Gal(\bar{\mathbb{Q}}_p ...
4
votes
2answers
248 views

Generalization of Kummer isomorphism?

This is a question I asked on math.stackexchange without success. Let $p$ be a prime number and denote by $\mathbb{F}_p(1)$ the one dimensional vector space over $\mathbb{F}_p$ endowed with an action ...